The Pythagorean Theorem

Pythagoras was not the first in antiquity to know about the remarkable theorem that bears his name, but he was the first to formally prove it using deductive geometry and the first to actively ‘market’ it (using today’s terms) throughout the ancient world. One of the earliest indicators showing knowledge of the relationship between right triangles and side lengths is a hieroglyphic-style picture, of a knotted rope (связанная узлом верёвка) having twelve equally-spaced knots (узел).

The rope was shown in a context suggesting its use as a workman’s tool (рабочий инструмент) for creating right angles, done via (через, посредством) the fashioning (придание вида, формы) of a 3-4-5 right triangle. Thus, the Egyptians had a mechanical device for demonstrating the converse of the Pythagorean Theorem for the 3-4-5 special case.

Not only did the Egyptians know of specific instances (примеры, отдельные случаи) of the Pythagorean Theorem, but also the Babylonians and Chinese some 1000 years before Pythagoras definitively institutionalized (устанавливать на практике) the general result circa (приблизительно) 500 BC.

The Pythagorean Theorem is a central theorem which states: in any right triangle the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a ² + b ² = c ², where a and b are the lengths of the legs and c is the length of the hypotenuse. The Pythagorean Property is true for all right triangles. There exist several proofs of the Pythagorean Theorem. Let’s discuss one of them.

A rectangle encloses the basic right triangle as shown below. The three triangles comprising the rectangle are similar, allowing the unknown dimensions x, y, to be solved via similarity principles in terms of a, b, and c. Once we have x, y, and z in hand, the proof proceeds as a normal dissection (разбиение, рассечение).

A Rectangular Dissection Proof:

1) x/b=a/c=›x=ab/c, y/b=b/c=›y= b²/c & z/a=a/c=›z= a ²/c

2) A=c{ab/c}=ab

3) A=1/2{ab/c× b²/c+ab+ab/c × a ²/c}=ab/2

{ b²/ c ²+1+ a ²/ c ²}

4) ab=ab/2{ b²/ c ²+1+ a ²/ c ²} =›2ab=ab{ b²/ c ²+1+ a ²/ c ²} =›2={b²/ c ²+1+ +a ²/ c ²} =›1= b²/ c ²+ a ²/ c ² =› b²/ c ²+ a ²/ c ²=1 =› a ² + b ² = c ²

Unit 5

“... it is written in the language of mathematics,

and its characters are triangles, circles,

and other geometrical figures... ”

Galileo (speaking of understanding the universe)

Grammar: Complex Object. Complex Subject.